[Proof of Theorem REF ]
We note that the results in [1]} hold true if we take \(\mathcal {H}:=\mathcal {H}^\ell \) . A standard argument using Lemma REF  shows that \(\mathcal {J}_\infty ^\ell :\mathcal {H}^\ell \rightarrow \mathbb {R}\)  satisfies the Palais-Smale condition at every level \(c\in \mathbb {R}\) . As in [1]} we set
\(\mathcal {U}:=\lbrace \mathbf {u}\in \mathcal {H}^\ell :\Vert u_i\Vert =1\text{ \ for \ }i=1,\ldots ,\ell ,\text{ \ and there exists \ }\mathbf {t}\in (0,\infty )^\ell \text{ \ such that \ }\mathbf {t}\mathbf {u}\in \mathcal {N}_\infty ^\ell \rbrace ,\) 
